Permutations

Item permutations consist in the list of all possible arrangements and ordering of elements in any order.

Example: The three letters A,B,C can be shuffled (anagrams) in 6 ways: A,B,C B,A,C C,A,B A,C,B B,C,A C,B,A

Permutations should not be confused with combinations (for which the order has no influence) or with arrangements also called partial permutations (k-permutations of some elements).

The best-known method is the Heap algorithm (method used by this dCode's calculator).

Step 1 - for each item, fix it at the beginning

Step 2 - repeat step 1 with the remaining items

Permutations can thus be represented as a tree of permutations:

Counting permutations uses combinatorics and factorials

Example: For $ n $ items, the number of permutations is equal to $ n! $ (factorial of $ n $)

Having a repeated item involves a division of the number of permutations by the number of permutations of these repeated items.

Example: DCODE 5 letters have $ 5! = 120 $ permutations but contain the letter D twice (these $ 2 $ letters D have $ 2! $ permutations), so divide the total number of permutations $ 5! $ by $ 2! $: $ 5!/2!=60 $ distinct permutations.

Permutations makes exponential values which needs huge computing servers with huge memory cells, so the generation must be paid.